Sequences and Series/Michaela P. Summary: Unit 3 is about sequences and series in math. Skills you will need before beginning this unit are: Order of Operations and Ratios. By the end of this unit you should be able to any specific term in any sequence using one of the shortcuts, figure out if a sequence is arithmetic or geometric, find the constant difference (if there is one), identify the rule or common ratio, solve for the sum of a sequence, and graph arithmetic sequences and geometric sequence. There are several pieces of key information that will be presented to you in this units. These pieces of information include the definitions of the words: sequence, recursive sequence, arithmetic sequence, geometric sequence, term, rule, common difference, and common ratio. You will also know all the different shortcuts, which will be listed at the bottom for you. This unit can be very easy, but it can also be very difficult if you are not careful. There are several mistakes you have to watch out for. For example, if you are looking for the sum of the series you have to know what the last number in the sequence is and what term that number is. Otherwise, you cannot solve your problem. Also, you have to look at several of the terms to figure about whether the sequence is arithmetic or geometric. Some geometric look like arithmetic sequence for the first two terms. You will also face several word problems in this unit. They look tricky, but they are actually quite simple, it is only a matter of identifying the the different terms. Here is an example problem: Julia makes $2.00 an hour for first hour of work, $4.00 her second hour, $6.00 her third hour and so on. How much money will she earn on her 12th hour of work? To solve this problem, you must first identify the first term and the common difference or ratio. In this problem the first term is 2 and the common difference is also 2. From here you can begin to use the shortcut to find a certain term in an arithmetic sequence. The term you are looking for here is t12. So you would put that at the front of the equation and insert 12 for n. Below is the solution and answer. t12=2+(12-1)2 t12=2+(11)2 t12=2+22 t12=24 Julia will earn $24.00 on her 12th hour of work. Shortcuts: Arithmetic Shortcut: tn=t1+(n-1)d Geometric Shortcut: tn=t1*r(n-1) Sum Shortcut: Sn=(t1+tn)n/2
Created Problems:
What is ruleof this sequence: 4, 7, 10, 13….
rule: +3
Using the shortcut, find t17 of this sequence: 2, 7, 12, 17, 22, 27…
t17=2+(17-1)5
t17=2+(16)5 t17=2+80 t17=82
Using the shortcut, find t11 of this sequence: 2, 6, 18, 54…
t11=2*3(11-1)
t11=2*3(10) t11= 2*59049 t11=118098
Using the shortcut, find the sum of this sequence: 4, 11, 18, 25, 32… 207
Sn=(4+207)30/2
Sn=(211)30/2 Sn=6330/2 Sn=3165 Applications: Algebra 2: Given the first term and the common difference of an arithmetic sequence find the first five terms. t1= 28, d = 10 t2= 28+10(1)t3= 28+ 10(2)t4= 28+ 10(3)t5= 28+ 10(4) t2=38t3= 48t4= 58t5= 68 a. 28, 38, 48, 58, 68 This is a problem from an algebra 2 worksheet. First, you have to figure out the equation to get the next 4 terms. I did this by taking the first term, 28, and adding the common difference, 10, multiplied by the number of the term I was looking for, i.e. 1-5.
SAT/ACT: Find the sum of the first 9 terms of the sequence: 6, 10, 14… t9=6+(9-1)4S9=(6+38)9/2 t9=6+(8)4S9=(44)9/2 t9=6+32S9=396/2 t9=38S9=198 a. 198 This problem is from an SAT practice test. To find the sum of the sequence, you first have to find the 9th term in the sequence. To do that, you take the first term, 6, and add it to the number of the term you are looking for, 9, minus 1, and multiplied by the common difference, 4. Once you know what the 9th term is, 38, you can begin to solve for the sum. To find the sum, you add the first term, 6, and the last term, 38, multiply that by the number of terms, 9, and divide all of that by 2.
Real life: You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance the object will fall in 6 seconds? t6=16+(6-1)32 t6=16+(5)32a. 176 feet t6=16+160 t6=176 This is an example of how you can use series and sequences in real life. To find the answer to this problem, you first must identify the first term, 16, and the common difference, 32. Then, you subtract 1 from 6, multiply that by 32, and add that answer to 16. Once you find the answer, 176, you have to put the unit at the end of the number, so that it is clear what you are talking
Summary: Unit 3 is about sequences and series in math. Skills you will need before beginning this unit are: Order of Operations and Ratios. By the end of this unit you should be able to any specific term in any sequence using one of the shortcuts, figure out if a sequence is arithmetic or geometric, find the constant difference (if there is one), identify the rule or common ratio, solve for the sum of a sequence, and graph arithmetic sequences and geometric sequence. There are several pieces of key information that will be presented to you in this units. These pieces of information include the definitions of the words: sequence, recursive sequence, arithmetic sequence, geometric sequence, term, rule, common difference, and common ratio. You will also know all the different shortcuts, which will be listed at the bottom for you. This unit can be very easy, but it can also be very difficult if you are not careful. There are several mistakes you have to watch out for. For example, if you are looking for the sum of the series you have to know what the last number in the sequence is and what term that number is. Otherwise, you cannot solve your problem. Also, you have to look at several of the terms to figure about whether the sequence is arithmetic or geometric. Some geometric look like arithmetic sequence for the first two terms.
You will also face several word problems in this unit. They look tricky, but they are actually quite simple, it is only a matter of identifying the the different terms. Here is an example problem:
Julia makes $2.00 an hour for first hour of work, $4.00 her second hour, $6.00 her third hour and so on. How much money will she earn on her 12th hour of work?
To solve this problem, you must first identify the first term and the common difference or ratio. In this problem the first term is 2 and the common difference is also 2. From here you can begin to use the shortcut to find a certain term in an arithmetic sequence. The term you are looking for here is t12. So you would put that at the front of the equation and insert 12 for n. Below is the solution and answer.
t12=2+(12-1)2
t12=2+(11)2
t12=2+22
t12=24
Julia will earn $24.00 on her 12th hour of work.
Shortcuts:
Arithmetic Shortcut: tn=t1+(n-1)d
Geometric Shortcut: tn=t1*r(n-1)
Sum Shortcut: Sn=(t1+tn)n/2
Created Problems:
- What is ruleof this sequence: 4, 7, 10, 13….
- rule: +3
- Using the shortcut, find t17 of this sequence: 2, 7, 12, 17, 22, 27…
- t17=2+(17-1)5
t17=2+(16)5t17=2+80
t17=82
- Using the shortcut, find t11 of this sequence: 2, 6, 18, 54…
- t11=2*3(11-1)
t11=2*3(10)t11= 2*59049
t11=118098
- Using the shortcut, find the sum of this sequence: 4, 11, 18, 25, 32… 207
- Sn=(4+207)30/2
Sn=(211)30/2Sn=6330/2
Sn=3165
Applications:
Algebra 2:
Given the first term and the common difference of an arithmetic sequence find the first five terms.
t1= 28, d = 10
t2= 28+10(1)t3= 28+ 10(2)t4= 28+ 10(3)t5= 28+ 10(4)
t2=38t3= 48t4= 58t5= 68
a. 28, 38, 48, 58, 68
This is a problem from an algebra 2 worksheet. First, you have to figure out the equation to get the next 4 terms. I did this by taking the first term, 28, and adding the common difference, 10, multiplied by the number of the term I was looking for, i.e. 1-5.
SAT/ACT:
Find the sum of the first 9 terms of the sequence: 6, 10, 14…
t9=6+(9-1)4S9=(6+38)9/2
t9=6+(8)4S9=(44)9/2
t9=6+32S9=396/2
t9=38S9=198
a. 198
This problem is from an SAT practice test. To find the sum of the sequence, you first have to find the 9th term in the sequence. To do that, you take the first term, 6, and add it to the number of the term you are looking for, 9, minus 1, and multiplied by the common difference, 4. Once you know what the 9th term is, 38, you can begin to solve for the sum. To find the sum, you add the first term, 6, and the last term, 38, multiply that by the number of terms, 9, and divide all of that by 2.
Real life:
You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance the object will fall in 6 seconds?
t6=16+(6-1)32
t6=16+(5)32a. 176 feet
t6=16+160
t6=176
This is an example of how you can use series and sequences in real life. To find the answer to this problem, you first must identify the first term, 16, and the common difference, 32. Then, you subtract 1 from 6, multiply that by 32, and add that answer to 16. Once you find the answer, 176, you have to put the unit at the end of the number, so that it is clear what you are talking